Question

in the given figure if radius of circle is 3 cm find the perimeter of triangle ABC​

Answer 1

In triangle BOC

let D be mid point of BC

so in triangle ODC (right triangle since BC is tangent)

OC = hypotenuse = 3√5 cm

OD = radius = 3 cm (given)

so DC^2 = (3√5)^2-3^2 = 45 - 9 = 36 (pythagoras theorem)

DC = 6 cm

so BC = 2DC = 12 cm

now the triangle ABC is equilateral since

<BOC = 360/3 = 120

in triangle BOC

<OCB = <OBC (since OB = OC)

 <OCB+<OBC= 180-<BOC

= 180-120

= 60

hence <OCB = <OBC = 30

now

<ACO = <OCB since OC is angle bisector

so

<ACO = 30

now <ACO + <OCB = 60 = <C

similarly <B = 60

therefore <A = 60

hence its an equilateral triangle with side BC = 12cm

hence perimeter is 12+12+12 = 36 cm ans